Isometries of 3-dimensional semi-Riemannian Lie groups
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915471972892672 |
|---|---|
| author | Chaib, Salah Ferreira, Ana Cristina Zeghib, Abdelghani |
| author_facet | Chaib, Salah Ferreira, Ana Cristina Zeghib, Abdelghani |
| contents | Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. By definition, the isometry group $\mathrm{Isom}(G, g)$ contains $G$ itself, acting by left translations. It turns out that, generically, $\mathrm{Isom}(G, g)$ is actually equal to $G$, and the natural question then becomes to classify those special metrics for which this is not the case. Using Lie-theoretical methods, we present a unified approach to obtain all pairs $(G, g)$ whose full isometry group $\mathrm{Isom}(G, g)$ has dimension greater than or equal to four. As a consequence, we determine, for every pair $(G, g)$, up to automorphism and scaling, the dimension of $\mathrm{Isom}(G, g)$, which can be three, four, or six. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isometries of 3-dimensional semi-Riemannian Lie groups Chaib, Salah Ferreira, Ana Cristina Zeghib, Abdelghani Differential Geometry Primary 53C30, Secondary 53C50, 53C20, 57S15, 57S20 Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. By definition, the isometry group $\mathrm{Isom}(G, g)$ contains $G$ itself, acting by left translations. It turns out that, generically, $\mathrm{Isom}(G, g)$ is actually equal to $G$, and the natural question then becomes to classify those special metrics for which this is not the case. Using Lie-theoretical methods, we present a unified approach to obtain all pairs $(G, g)$ whose full isometry group $\mathrm{Isom}(G, g)$ has dimension greater than or equal to four. As a consequence, we determine, for every pair $(G, g)$, up to automorphism and scaling, the dimension of $\mathrm{Isom}(G, g)$, which can be three, four, or six. |
| title | Isometries of 3-dimensional semi-Riemannian Lie groups |
| topic | Differential Geometry Primary 53C30, Secondary 53C50, 53C20, 57S15, 57S20 |
| url | https://arxiv.org/abs/2502.17193 |